You have two questions in one: the first one is the full-time student status, the second one is the mid-August time frame. 

To answer your first question - Yes, you are considered to be full-time if the school's form 1098-T has the box checked that you were "at least half-time" student. If you don't have the form, ask the school if they consider you a full-time student, when you start doesn't matter; your course load does. 

Your second question - the IRS says that you must be a full-time student for any part of 5 months" during the tax year so your time frame does qualify an education credit.